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Artificial Neural Networks
Artificial Neural Networks• Interconnected networks of simple units
("artificial neurons"). – Weight wij is the weight of the ith input into
unit j. – Depending on the final output value we
determine the class. – If more than one output units then we choose
the one with the greatest value.
• Learning takes place by adjusting the weights in the network– so that the desired output is produced
whenever a training instance is presented.
Single Perceptron Unit• We start by looking at a
simpler kind of "neural-like" unit called a perceptron.
0or
0
isequation hyperplane then the
and 1let
0
00
n
jjj xw
bwx
xw
)()( xwx signh
Depending on the value of h(x) it outputs one class or the other.
Beyond Linear Separability• Values of the XOR boolean
function cannot be separated by a single perceptron unit.
Multi-Layer Perceptron• Solution: Combine multiple
linear separators.
• Introduction of "hidden" units into NN make them much more powerful: – they are no longer limited to
linearly separable problems.
• Earlier layers transform the problem into more tractable problems for the latter layers.
Example: XOR problem
Output: “class 0”
or “class 1”
Example: XOR problem
Example: XOR problem
w23o2+w13o1+w03=0w03=-1/2, w13=-1, w23=1
o2-o1-1/2=0
Multi-Layer Perceptron• Any set of training points can be separated by a three-layer
perceptron network.
• “Almost any” set of points is separable by a two-layer perceptron network.
Backpropagation techniqueHigh level summary:
1. Present a training sample to the neural network.
2. Calculate the error in each output neuron. This is the local error.
3. Adjust the weights of each neuron to lower the local error.
4. Assign "blame" for the local error to neurons at the previous level, giving greater responsibility to neurons connected by stronger weights.
5. Repeat from step 3 on the neurons at the previous level, using each one's "blame" as its error.
Autonomous Land Vehicle In a Neural Network (ALVINN)
• ALVINN is an automatic steering system for a car based on input from a camera mounted on the vehicle. – Successfully demonstrated in a cross-country trip.
ALVINN• The ALVINN neural
network is shown here. It has – 960 inputs (a 30x32
array derived from the pixels of an image),
– 4 hidden units and
– 30 output units (each representing a steering command).
SVMs vs. ANNs• Comparable in practice.
• Some comment:
"SVMs have been developed in the reverse order to the development of neural networks (NNs). SVMs evolved from the sound theory to the implementation and experiments, while the NNs followed more heuristic path, from applications and extensive experimentation to the theory.“ (Wang 2005)
Soft Threshold• A natural question to ask is whether we could use gradient
ascent/descent to train a multi-layer perceptron.
• The answer is that we can't as long as the output is discontinuous with respect to changes in the inputs and the weights. – In a perceptron unit it doesn't matter how far a point is from the
decision boundary, we will still get a 0 or a 1.
• We need a smooth output (as a function of changes in the network weights) if we're to do gradient descent.
Sigmoid Unit• Commonly used in neural nets is a
"sigmoid" (S-like) function (see on the right).
– The one used here is called the logistic function.
• Value z is also called the "activation" of a neuron.
Training• Key property of the sigmoid is that it is differentiable.
– This means that we can use gradient based methods of minimization for training.
• The output of a multi-layer net of sigmoid units is a function of two vectors, the inputs (x) and the weights (w). – Well, as we train the ANN the training instances are considered
fixed.
• The output of this function (y) varies smoothly with changes in the weights.
Training
Training
½ is only to simplify the derivations.
Gradient Descent
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w
y
w
yy
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yyE
n
m
mmm
mm
w
w
ww
ww
w
wx
wxwx
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follows as change We
,...,),( where
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)),((2
1
1
2
We follow gradient descent
Gradient of the training error is computed as a function of the weights.
Online version: We consider each time only the error for one data item
As a shorthand, we will denote y(xm,w) just by y.
Gradient Descent – Single Unit
mi
i
ii
z
n
i
mii
n
m
xz
zs
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z
z
zs
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,...,),(
0
1
wxw
mii
m
mi
mii
xw
z
zsyy
z
E
xz
zsyyww
)()(
)()(
Delta rule
Substituting in the equation of previous slide we get (for the arbitrary ith element of w):
Derivative of the sigmoid
)1(
))(1)((
11
1
]][)1([
)1()(
1
1)(
2
1
yy
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e
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z
mi
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m
mi
mmii
xyyyy
xyyyy
xz
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))(1(
)1()(
)(
)(
Generalized Delta RuleFor an output unit p we similarly have:
mi
mppp
mipip
yyyyy
yw
))(1(
y3
z3
y1
z1
y2
z2
1 2
3
1 1x1x2
w01
w03
w02w11 w22
w21
w13 w23
w12
1p=3 in this example
Backpropagation Example
y3
z3
y1
z1
y2
z2
1 2
3
1 1x1x2
w01
w03
w02w11 w22
w21
w13 w23
w12
1
133111
233222
3333
)1(
)1(
))(1(
wyy
wyy
yyyy m
First do forward propagation:Compute zi’s and yi’s.
232323
131313
30303 )1(
yww
yww
ww
222222
121212
20202 )1(
xww
xww
ww
212121
111111
10101 )1(
xww
xww
ww
We'll see soon why delta2 and delta3 have these formulas.
Deriving 2 and 2
y3
z3
y1
z1
y2
z2
1 2
3
1 1x1x2
w01
w03
w02w11 w22
w21
w13 w23
w12
1
)1(
))(1)((
)(
)(
22233
22233
2
2233
2
2233
2
2231133
2
33
2
3
3
22
yyw
zszsw
z
zsw
z
yw
z
ywyw
z
z
z
z
z
E
z
E
We similarly derive delta1.
Backpropagation Algorithm1. Initialize weights to small random values2. Choose a random sample training item, say (xm, ym)
3. Compute total input zj and output yj for each unit (forward prop)
4. Compute p for output layer p = yp(1-yp)(yp-ym)
5. Compute j for all preceding layers by backprop rule6. Compute weight change by descent rule (repeat for all weights)
• Note that each expression involves data local to a particular unit, we don't have to look around summing things over the whole network.
• It is for this reason, simplicity, locality and, therefore, efficiency that backpropagation has become the dominant paradigm for training neural nets.
Generalized Delta Rule
ijijij
jdownstreamkjkkjjj
yww
wyy
)(
)1(
In general, for a hidden unit j we have
Input And Output Encoding• For neural networks, all attribute values must be encoded in a standardized
manner, taking values between 0 and 1, even for categorical variables.
• For continuous variables, we simply apply the min-max normalization:X* = [X - min(X)]/[max(X)-min(X)]
• For categorical variables use indicator (flag) variables. – E.g. marital status attribute, containing values single, married, divorced.
– Records for single would have
1 for single, and 0 for the rest, i.e. (1,0,0)
– Records for married would have
1 for married, and 0 for the rest, i.e. (0,1,0)
– Records for divorced would have
1 for divorced, and 0 for the rest, i.e. (0,0,1)
– Records for unknown would have
0 for all, i.e. (0,0,0)
• In general, categorical attributes with k values can be translated into k - 1 indicator attributes.
Output• Neural network output nodes always return a continuous value between 0 and
1 as output.
• Many classification problems have a dichotomous result, with only two possible outcomes. – E.g., “Meningitis, yes or not"
• For such problems, one option is to use a single output node, with a threshold value set a priori which would separate the classes.– For example, with the threshold of “Yes if output 0.3," an output of 0.4 from
the output node would classify that record as likely to be “Yes”.
• Single output nodes may also be used when the classes are clearly ordered. E.g., suppose that we would like to classify patients’ disease levels. We can say:– If 0 output < 0.33, classify “mild”– If 0.33 output < 0.66, classify “severe”– If 0.66 output < 1, classify “grave”
Multiple Output Nodes• If we have unordered categories for the target attribute, we
create one output node for each possible category. – E.g. for marital status as target attribute, the network would have
four output nodes in the output layer, one for each of: • single, married, divorced, and unknown.
• Output node with the highest value is then chosen as the classification for that particular record.
NN for Estimation And Prediction• Since NN produce continuous output, they can be used for
estimation and prediction.
• Suppose, we are interested in predicting the price of a stock three months in the future. – Presumably, we would have encoded price information using the
min-max normalization.
– However, the neural network would output a value between zero and 1.
• The min-max normalization needs to be inverted.
• This denormalization is:prediction = output * (max – min) + min
ANN Example
y3
z3
y1
z1
y2
z2
1 2
3
1 1x1
x3
w01
w03
w02w11 w22
w21
w13 w23
w12
1
x2
w32
w31
Learning WeightsFor an output unit p we similarly have:
mi
mppp
mipip
yyyyy
yw
))(1(
p=3 in this exampley3
z3
y1
z1
y2
z2
1 2
3
1 1x1
x3
w01
w03
w02w11 w22
w21
w13 w23
w12
1
x2
w32
w31
Backpropagation
133111
233222
3333
)1(
)1(
))(1(
wyy
wyy
yyyy m
First do forward propagation:Compute zi’s and yi’s.
232323
131313
30303 )1(
yww
yww
ww
323232
222222
121212
20202 )1(
xww
xww
xww
ww
313131
212121
111111
10101 )1(
xww
xww
xww
ww
y3
z3
y1
z1
y2
z2
1 2
3
1 1x1
x3
w01
w03
w02w11 w22
w21
w13 w23
w12
1
x2
w32
w31
Backpropagation ExampleFirst do forward propagation:Compute zi’s and yi’s.
Suppose we have initially chosen (randomly) the weights given in the table.
Also, in the table is given one training instance (first column).
x0 = 1.0 w01 = 0.5 w02 = 0.7 w03 = 0.5
x1 = 0.4 w11 = 0.6 w12 = 0.9 w13 = 0.9
x2 = 0.2 w21 = 0.8 w22 = 0.8 w23 = 0.9
x3 = 0.7 w31 = 0.6 w32 = 0.4
y3
z3
y1
z1
y2
z2
1 2
3
1 1x1
x3
w01
w03
w02w11 w22
w21
w13 w23
w12
1
x2
w32
w31
Feed-Forward Example
y3
z3
y1
z1
y2
z2
1 2
3
1 1x1
x3
w01
w03
w02w11 w22
w21
w13 w23
w12
1
z1 = 1.0*0.5+0.4*0.6+0.2*0.8+0.7*0.6 = 1.32y1 = 1/(1+e^(-z1)) = 1/(1+e(-1.32)) = 0.7892
z2 = 1.0*0.7+0.4*0.9+0.2*0.8+0.7*0.4 = 1.5y2 = 1/(1+e^(-z2)) = 1/(1+e(-1.5)) = 0.8175
z3 = 1.0*0.5+ 0.79*0.9+ 0.82*0.9= 1.95y3 = 1/(1+e^(-z3)) = 1/(1+e(-1.95)) = 0.87
x0 = 1.0 w01 = 0.5 w02 = 0.7 w03 = 0.5
x1 = 0.4 w11 = 0.6 w12 = 0.9 w13 = 0.9
x2 = 0.2 w21 = 0.8 w22 = 0.8 w23 = 0.9
x3 = 0.7 w31 = 0.6 w32 = 0.4
x2
w32
w31
Backpropagation• So, the network output, for the given training example, is
y3=0.87.
• Assume the actual value of the target attribute is y=0.8
• Then the prediction error equals 0.8 – 0.8750 = -0.075.
Now 3 = y3(1-y3)(y3-y) = 0.87*(1-0.87)*(0.87-0.8) = 0.008
• Let’s have a learning rate of =0.01. Then, we update weights:w03 = w03 - 3 (1) = 0.5 - 0.01*0.008*1 = 0.49918
w13 = w13 - 3 y1 = 0.9 - 0.01*0.008* 0.7892 = 0.8999
w23 = w23 - 3 y2 = 0.9 - 0.01*0.008* 0.8175 = 0.8999
Backpropagation 2 =
y2(1-y2)3w23 = 0.8175*(1-0.8175)*0.008*0.9 = 0.001
1 =
y1(1-y1)3w13 = 0.7892*(1- 0.7892)*0.008*0.9 = 0.0012
• Then, we update weights:
w02 = w02 - 2 (1) = 0.7 - 0.01*0.001*1 = 0.6999
w12 = w12 - 2 x1 = 0.9 - 0.01*0.001* 0.4 = 0.8999
w22 = w22 - 2 x2 = 0.8 - 0.01*0.001* 0.2 = 0.7999
w32 = w32 - 2 x3 = 0.4 - 0.01*0.001* 0.7 = 0.3999
w01 = w01 - 1 (1) = 0.5 - 0.01*0.001*1 = 0.4999
w11 = w11 - 1 x1 = 0.6 - 0.01*0.001* 0.4 = 0.5999
w21 = w21 - 1 x2 = 0.8 - 0.01*0.001* 0.2 = 0.7999
w31 = w31 - 1 x3 = 0.6 - 0.01*0.001* 0.7 = 0.5999